Existential Instantiation, Arbitrary Reference and Supposition Wylie Breckenridge Existential instantiation is a rule of inference that allows us infer, from the proposition that there are some p things, the proposition that a is a p thing. What role does 'a' play here? According to one account, recently defended by Breckenridge and Magidor, we use 'a' to refer to a p thing. I argue that this cannot be right. I propose an alternative account, according to which we use 'a' to refer to a supposedly p thing. 1. Existential instantiation Suppose that p and q are properties. Here is a valid argument: 1. There are some p things 2. All p things are q things Therefore, C. There are some q things If its validity is not immediately apparent then we can make it more apparent by adding some intermediate steps, as follows: 1. There are some p things (Premise) 2. All p things are q things (Premise) 3. a is a p thing (From 1, by existential instantiation) 4. a is a q thing (From 2 and 3, by universal instantiation) C. There are some q things (From 4, by existential generalisation) Line 3 is an application of existential instantiation. When presenting this argument we might more naturally say, “Let a be p thing”, but it is standardly presented in writing as “a is a p thing”. 2. The role of ‘a’ and the p thing account What role does ‘a’ play in line 3, and then again in line 4? Although existential instantiation is discussed in standard textbooks on first-order logic, typically not much is said about the role of ‘a’, other than that it ‘stands for an arbitrary p thing’ (or something like that). Proving that the rules involving the introduction and elimination of terms such as ‘a’ are sound and complete tells us that by following these rules we can prove exactly what we should be able to prove, but it does not tell us much about what ‘a’ means. This turns out to be complicated. Various accounts have been offered. According to some accounts we use ‘a’ as a meaningless symbol. According to others we use it as a variable. According to Kit Fine (1985) we use it to refer to the arbitrary p thing – a special kind of thing that is not an ordinary p thing. (For details and further references see B&M 2012.) Breckenridge and Magidor (2012) have recently argued against each of these accounts. They defend an alternative referential account, according to which we use ‘a’ to refer 1 to a p thing (an ordinary p thing, not Fine’s arbitrary p thing). On this view line 3 actually serves a dual role – we use it to fix the reference of ‘a’ to a p thing, and also to express the proposition that a is a p thing. By writing “a is a p thing” we emphasize the second of these two roles; we could instead emphasize the first role by writing “Let a be a p thing”. Defending this view of ‘a’ is not the main aim of B&M’s paper – their main aim is to defend what they call ‘arbitrary reference’, the view that we can refer to things arbitrarily. Nevertheless, their main argument for arbitrary reference depends on this view of the role of ‘a’. They argue: In line 3 we get ‘a’ to refer to a p thing; we can only get ‘a’ to refer to a p thing if we can arbitrarily refer; therefore, we can arbitrarily refer. 3. Problems for the p thing account I am persuaded by B&M’s arguments against meaningless symbol accounts of ‘a’, against variable accounts of ‘a’, and against Fine’s referential account of ‘a’, and for the rest of this paper I will assume that these accounts are wrong and that some other referential account is right. But I’m not persuaded by the referential account defended by B&M. In fact, I think it must be wrong. B&M do not say such much about what kind of p thing we get ‘a’ to refer to in line 3, and it is difficult to see what kind of p thing it could be. It cannot be an actual p thing, because for some properties p there are no actual p things. Nor can it be a possible p thing, because for some properties p there are no possible p things (it might also be impossible to refer to merely possible p things, at least for some properties p). So they can’t claim that we get ‘a’ to refer to an actual or possible p thing. It won’t do for them to say that these properties are special – that for these properties either a different account is needed or that existential instantiation does not work. The original argument is valid, and the extended argument helps to show that it is valid, no matter which properties p and q are; in particular, no matter which property p is. So they need an account of ‘a’ that works for all properties p. There are also problems for the idea that ‘a’ refers to a p thing even for some properties p that have actual instances. First, let p be the property of being male. This has actual instances. But if q is a property that is inconsistent with p, such as being not male, then there are no possible circumstances in which the premises are both true. So, if ‘a’ is supposed to refer to some p thing in some circumstance is which the premises are both true then ‘a’ cannot refer to anything (because there are no such circumstances). Second, let p be the property of being a never-referred-to thing (see Haze (Forthcoming)). This has actual instances. But ‘a’ cannot refer to any of them, because that amounts to ‘a’ referring to a never-referred-to thing, which it cannot do, because if it did then that thing would not be a never-referred-to thing. 2 Again, it won’t do for B&M to say that these properties are special. The original argument is valid, and the extended argument helps to show that it is valid, no matter which properties p and q are. They need an account that works for all properties p and q. So I don’t see how a referential account of ‘a’ can successfully maintain that we get ‘a’ to refer to a p thing. 4. A proposal: the supposedly p thing account I have a suggestion, and presenting this suggestion is the main purpose of this paper. A referential account of ‘a’ cannot claim that we use ‘a’ to refer to a p thing (as we have just seen). But it might be able to claim that we use ‘a’ to refer to a supposedly p thing (that is, a thing that supposedly has p). Moreover, this might still give B&M an argument for arbitrary reference, as follows: In line 3 we get ‘a’ to refer to a supposedly p thing; we can only get ‘a’ to refer to a supposedly p thing if we can arbitrarily refer; therefore, we can arbitrarily refer. And it might allow them to avoid the problems that I have presented above. My task now is to explain all of this. Suppose that we are developing a referential account of the role of ‘a’ in the extended argument – that is, an account according to which we use ‘a’ to refer to something. Let’s think through how the account should go. One thing that we’d like the account to do is explain how the extended argument helps us to see that the original argument is valid. How might our account do this? We might try saying this: by adding lines 3 and 4 to the argument we break the inference down into some smaller steps, each of which is more clearly valid than the step from 1 and 2 to C. (i.e. lines 3 and 4 are intermediate consequences.) The idea would be this: C follows from 1 and 2, because C follows from 4, and 4 follows from 2 and 3, and 3 follows from 1. This is a common way of adding lines to an argument to make its validity more apparent. But we can’t say this. In particular, we can’t say that 3 follows from 1 (although we might be able to say that 4 follows from 2 and 3). According to the account that we are developing, ‘a’ refers to something. For 3 to follow from 1 we need the following to be the case: it is not possible for 1 to be true while 3 is false; that is, it is not possible for there to be some p things without a being a p thing. So we need ‘a’ to refer to something x such that: necessarily, if there are any p things then x is a p thing. For some properties p there is such a thing. The property of being Bill Gates is one such property (necessarily, if there are any things that have this property then Bill Gates has this property). So is the property of being a number (necessarily, if there are any things that have this property then 2 has this property), and the property of being even (necessarily, if there are any things that have this property then 2 has this property). 3 But, and this is the problem, for some properties p there is no such thing.